An Efficient Method to Solve Thermal Wave Equation

Mohamed Salah, R. M. Amer, M. S. Matbuly

Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig, Egypt.

DOI: 10.4236/am.2014.53052                

Abstract

In this paper, an efficient technique of differential quadrature method and perturbation method is employed to analyze reaction-diffusion problems. An efficient method is presented to solve thermal wave propagation model in one and two dimensions. The proposed method marches in the time direction block by block and there are several time levels in each block. The global method of differential quadrature is applied in each block to discretize both the spatial and temporal derivatives. Furthermore, the proposed method is validated by comparing the obtained results with the available analytical ones and also compared with the hybrid technique of differential quadrature method and Runge-Kutta fourth order method.

Keywords

Reaction-Diffusion; Block Marching; Differential Quadrature; Perturbation

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	J. Opsal, A. Rosencwaig and D. L.Willenborg, “Thermal-Wave Detection and Thin-Film Thickness Measurements with Laser Beam Deflection,” Applied Optics, Vol. 22, No. 20, 1983, pp. 3169-3176. http://dx.doi.org/10.1364/AO.22.003169
[2]	Z. Y. Yan, “Generalized Method and Its Application in the Higher-Order Nonlinear Schrodinger Equation in Nonlinear Optical Fibres,” Chaos Solitons and Fractals, Vol. 16, No. 5, 2003, pp. 759-766. http://dx.doi.org/10.1016/S0960-0779(02)00435-6
[3]	J. Mei, H. Zhang and D. Jiang, “New Exact Solutions for a Reaction-Diffusion Equation and a Quasi-Camassa Holm Equation,” Applied Mathematics E-Notes, Vol. 4, 2004, pp. 85-91.
[4]	E. S. Fahmy and H. A. Abdusalam, “Exact Solutions for Some Reaction Diffusion Systems with Nonlinear Reaction Polynomial Terms,” Applied Mathematical Sciences, Vol. 3, No. 11, 2009, pp. 533-540.
[5]	M. S. H. Chowdhury and I. Hashim, “Analytical Solution for Cauchy Reaction-Diffusion Problems by Homotopy Perturbation Method,” Sains Malaysiana, Vol. 39, No. 3, 2010, pp. 495-504.
[6]	S. Puri and K. Wiese, “Perturbative Linearization of Reaction-Diffusion Equations,” Journal of Physics A: Mathematical and General, Vol. 36, 2003, pp. 2043-2054. http://dx.doi.org/10.1088/0305-4470/36/8/303
[7]	L. D. Ropp, N. J. Shadid and C. C. Ober, “Studies of the Accuracy of Time Integration Methods for Reaction-Diffusion Equations,” Journal of Computational Physics, Vol. 194, No. 2, 2004, pp. 544-574. http://dx.doi.org/10.1016/j.jcp.2003.08.033
[8]	R. G. Marcus, “Finite-Difference Schemes for Reaction-Diffusion Equations Modeling Predator—Prey Interactions in MATLAB,” Bulletin of Mathematical Biology, Vol. 69, No. 3, 2007, pp. 931-956. http://dx.doi.org/10.1007/s11538-006-9062-3
[9]	B. Liu, M. B. Allen, H. Kojouharov and B. Chen, “Finite-Element Solution of Reaction-Diffusion Equations with Advection,” Computational Mechanics, 1996, pp. 3-12.
[10]	X. Christos and L. Oberbroeckling, “On the Finite Element Approximation of Systems of Reaction—Diffusion Equations by p/hp Methods,” Global Science, Vol. 28, No. 3, 2010, pp. 386-400.
[11]	G. Meral and M. S. Tezer, “Solution of Nonlinear Reaction-Diffusion Equation by Using Dual Reciprocity Boundary Element Method with Finite Difference or Least Squares Method,” Advances in Boundary Element Techniques, 2008, pp. 317-22.
[12]	C. Shu, “Differential Quadrature and Its Application in Engineering,” Springer Verlag, London, 2000. http://dx.doi.org/10.1007/978-1-4471-0407-0
[13]	T. Y. Wu and G. R. Liu, “A Differential Quadrature as a Numerical Method to Solve Differential Equations,” Computational Mechanics, Vol. 24, No. 3, 1999, pp. 197-205. http://dx.doi.org/10.1007/s004660050452
[14]	G. Meral, “Solution of Density Dependent Nonlinear Reaction-Diffusion Equation Using Differential Quadrature Method,” World Academy of Science, Engineering and Technology, Vol. 41, 2010, pp. 1178-1183.
[15]	V. Kajal, “Numerical Solutions of Some Reaction-Diffusion Equations by Differential Quadrature Method,” International Journal of Applied Mathematics and Mechanics, Vol. 6, No. 14, 2010, pp. 68-80.
[16]	M. Salah, R. M. Amer and M. S. Matbuly, “The Differential Quadrature Solution of Reaction-Diffusion Equation Using Explicit and Implicit Numerical Schemes,” Applied Mathematics, 2014.
[17]	C. Shu, Q. Yao and K. S. Yeo, “Block-Marching in Time with DQ Discretization: An Efficient Method for Time-Dependent Problems,” Computer Methods in Applied Mechanics and Engineering, Vol. 191, No. 41, 2002, pp. 4587-4579. http://dx.doi.org/10.1016/S0045-7825(02)00387-0
[18]	J. S. Nadjafi and A. Ghorbani, “He’s Homotopy Perturbation Method: An Effective Tool for Solving Nonlinear Integral and Integro-Differential Equations,” Computers and Mathematics with Applications, Vol. 58, No. 11-12, 2009, pp. 2379-2390. http://dx.doi.org/10.1016/j.camwa.2009.03.032
[19]	H. S. Prasad and Y. N. Reddy, “Numerical Solution of Singularly Perturbed Differential-Difference Equations with Small Shifts of Mixed Type by Differential Quadrature Method,” American Journal of Computational and Applied Mathematics, Vol. 2, No. 1, 2012, pp. 46-52. http://dx.doi.org/10.5923/j.ajcam.20120201.09
[20]	J. P. Hambleton and S. W. Sloan, “A Perturbation Method for Optimization of Rigid Block Mechanisms in the Kinematic Method of Limit Analysis,” Computers and Geotechnics, Vol. 48, 2013, pp. 260-271. http://dx.doi.org/10.1016/j.compgeo.2012.07.012
[21]	M. Bastani and D. K. Salkuyeh, “A Highly Accurate Method to Solve Fisher’s Equation,” Indian Academy of Sciences, Vol. 78, No. 3, 2012, pp. 335-346.
[22]	W. Y. Yang, W. Cao, T. Chung, J. Morris, et al., “Applied Numerical Methods Using Matlab,” John Wiley & Sons, Hoboken, New Jersey, 2005. http://dx.doi.org/10.1002/0471705195
[23]	E. Kreyszig, “Advanced Engineering Mathematics,” John Wiley & Sons, Columbus, 2006.